Generalized Linear Least Squares (LSE and GLM) Problems

Driver routines are provided for two types of generalized linear least squares problems.

The first is

$$\min _{x} \Vert c - Ax\Vert _2 \;\;\; \mbox{subject to} \;\;\; B x = d$$ (2.2)

where A is an m-by-n matrix and B is a p-by-n matrix, c is a given m-vector, and d is a given p-vector, with $p \leq n \leq m+p$. This is called a linear equality-constrained least squares problem (LSE). The routine xGGLSE solves this problem using the generalized RQ (GRQ) factorization, on the assumptions that B has full row rank p and the matrix $\left( \begin{array}{c} A \\ B \end{array} \right)$ has full column rank n. Under these assumptions, the problem LSE has a unique solution.

The second generalized linear least squares problem is

$$\min _{x} \Vert y\Vert _2 \;\;\; \mbox{subject to} \;\;\; d = A x + B y$$ (2.3)

where A is an n-by-m matrix, B is an n-by-p matrix, and d is a given n-vector, with $m \leq n \leq m+p$. This is sometimes called a general (Gauss-Markov) linear model problem (GLM). When B = I, the problem reduces to an ordinary linear least squares problem. When B is square and nonsingular, the GLM problem is equivalent to the weighted linear least squares problem:

$$\min_x \Vert B^{-1}(d-Ax) \Vert _2$$

The routine xGGGLM solves this problem using the generalized QR (GQR) factorization, on the assumptions that A has full column rank m, and the matrix ( A, B ) has full row rank n. Under these assumptions, the problem is always consistent, and there are unique solutions x and y. The driver routines for generalized linear least squares problems are listed in Table 2.4.

Table 2.4: Driver routines for generalized linear least squares problems

Operation	Single precision		Double precision
	real	complex	real	complex
solve LSE problem using GRQ	SGGLSE	CGGLSE	DGGLSE	ZGGLSE
solve GLM problem using GQR	SGGGLM	CGGGLM	DGGGLM	ZGGGLM